I am looking for a good reference on the measure theoretic construction of the spaces $L^p(M;E)$ for a vector bundle $\pi \colon E \rightarrow M$ on a Riemannian manifold $(M,g)$, by which I mean starting from measurable sections and not simply completing with respect to the $L^p$-norms. Specifically, I am also very interested in a reference to the claim that these spaces are isometrically isomorphic to $L^p(M;\mathbb{R}^n)$, as proved in https://math.stackexchange.com/a/4141964/636279.
Thanks a lot!